You said "How do I see what effect the two DV's have combined". So, I believe your two DV's are somehow related based on theory. In this case, running independent equations each for one DV is totally wrong! Imagine you want to measure "speaking skill" and one of your DV's is "accuracy of talking" and the other one is "fluency of talking". In this respect, theory says that fluency and accuracy are related and are two sub-dimensions of the same concept-speaking skill. In this case, the two DV's should be considered together. Whenever it is the case, you have a continuous DV and categorical IV(s), Multivariate ANOVA (MANOVA) should be used. MANOVA is a variant of ANOVA that can incorporate multiple continuous DV's. Also, the number of IV's is irrelevant here as both ANOVA and MANOVA can accommodate 1 IV or 2 IV's or 3 IV's, etc. (called one way, two way, three-way etc. ANOVA or MANOVA)
Also, pay attention that MANOVA is a parametric test, meaning that your measurement must be continuous. it is hard to assume that measurement by any Likert-type scale with less than 7 anchors is continuous.